Fellaini appeared to butt Potters defender Ryan Shawcross and the Belgian could now face a ban if he is charged by the FA after referee Mark Halsey missed the incident.
Everton manager David Moyes took a dim view of Fellaini's actions and said after the game: "I have told him in the dressing room that it is not acceptable. I won't accept it as a manager that he does that."
Fellaini later issued a statement on the club's official website which read: "I apologise completely to Ryan Shawcross, my team mates and to our fans at the game.
"There was a lot of pushing and pulling going on inside the Stoke penalty area and I didn't feel I was getting any protection from the officials.
"Nevertheless I know I shouldn't have done what I did.
"I have also apologised to the manager and the staff. I have no excuses, I was disappointed with the way I was being treated and I lost my temper, which was unprofessional of me."
Despite Fellaini's apology, the incident is likely to come under scrutiny from the Football Association's disciplinary panel when they review the footage this week.
Moyes said after the game that Everton would accept any punishment, with referee Halsey's failure to act allowing the FA scope to hand down retrospective punishment.
Moyes admitted he thought Fellaini had warranted a red card and emphasised he would not stand for that sort of behaviour from his players.
"We will take the punishment if it is given for it," Moyes said.
Asked how he might compensate for losing a player as influential as Fellaini to suspension should the 25-year-old be handed a ban by the FA, Moyes said: "There is no absence yet, but he is a really good player and a big part of the team, and that is why I think the biggest disappointment, if anything does happen, is that he will have let the team down."
He added: "I only normally complain if I think we are getting hard done by and I have said a few things about how I think we have been treated by decisions recently.
"So I think if you are going to talk about it with your own team... Today, it should have been a sending-off in the game.
"If the referee (Mark Halsey) had been able to see it, it should have been."
Stoke manager Tony Pulis claimed Fellaini had in fact been guilty of three unpunished offences this afternoon, and referenced others he felt had been committed by different players against his team this season.
"I'll say this - David Luiz's challenge on Jon Walters (in a 1-0 defeat at Chelsea), Ciaran Clark's challenge on Glenn Whelan last week (in a 0-0 draw at Aston Villa), and then the three incidents today," Pulis said. "I thought the rules were set out in stone.
"I like Mark Halsey and I think he's a good referee, but I don't know whether he has seen it.
"If he has seen it and not acted, then you really want to know why he has not acted.
"I don't think you can hide behind anything and what he (Fellaini) has done is wrong."